This work was devoted to a minimum-energy optimal control problem governed by a fourth-order reaction–diffusion equation in which the control acts through a drift term. Such bilinear control problems arise naturally in transport phenomena and higher-order diffusion models, but their analysis is challenging because of the nonlinear coupling between the state and the control, as well as the difficulty of characterizing controls that are truly optimal with respect to energy. To overcome this difficulty, we formulated a family of penalized optimal control problems and characterized the minimum energy control as the limit of their solutions. The analysis was based on the associated optimality system, involving the state and adjoint equations, together with tools from convex optimization. Under appropriate assumptions, we established the existence and uniqueness of the optimal control, which provides a well-posed framework for the problem under consideration. For the numerical approximation, the resulting optimality system was solved by means of a conjugate gradient algorithm that requires, at each iteration, only the solution of the state equation and the adjoint equation. Numerical simulations showed a clear decrease in the tracking error and indicated that the computed state approaches the desired configuration with a small relative error after convergence. These results confirmed the stability, accuracy, and efficiency of the proposed approach for the minimum-energy control of fourth-order advection–reaction–diffusion systems.
Citation: Mofareh Alhazmi, Maawiya Ould Sidi. Bilinear optimal control problem with minimum energy for a fourth-order reaction diffusion system[J]. AIMS Mathematics, 2026, 11(4): 10274-10310. doi: 10.3934/math.2026425
This work was devoted to a minimum-energy optimal control problem governed by a fourth-order reaction–diffusion equation in which the control acts through a drift term. Such bilinear control problems arise naturally in transport phenomena and higher-order diffusion models, but their analysis is challenging because of the nonlinear coupling between the state and the control, as well as the difficulty of characterizing controls that are truly optimal with respect to energy. To overcome this difficulty, we formulated a family of penalized optimal control problems and characterized the minimum energy control as the limit of their solutions. The analysis was based on the associated optimality system, involving the state and adjoint equations, together with tools from convex optimization. Under appropriate assumptions, we established the existence and uniqueness of the optimal control, which provides a well-posed framework for the problem under consideration. For the numerical approximation, the resulting optimality system was solved by means of a conjugate gradient algorithm that requires, at each iteration, only the solution of the state equation and the adjoint equation. Numerical simulations showed a clear decrease in the tracking error and indicated that the computed state approaches the desired configuration with a small relative error after convergence. These results confirmed the stability, accuracy, and efficiency of the proposed approach for the minimum-energy control of fourth-order advection–reaction–diffusion systems.
| [1] | J. L. Lions, Optimal control of systems governed by partial differential equations, Berlin: Springer, 170 (1971). |
| [2] |
A. Moussa, Some variants of the classical Aubin–Lions lemma, J. Evol. Equ., 16 (2016), 65–93. https://doi.org/10.1007/s00028-015-0293-3 doi: 10.1007/s00028-015-0293-3
|
| [3] | J. L. Lions, E. Magenes, Problèmes aux limites non-homogènes et applications, Paris: Dunod, 1 (1968). |
| [4] | J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Berlin: Springer, 1 (2012). |
| [5] | F. Tröltzsch, Optimal control of partial differential equations: Theory, methods and applications, Providence: American Mathematical Society, 112 (2010). |
| [6] | R. Glowinski, J. L. Lions, J. He, Exact and approximate controllability for distributed parameter systems: A numerical approach, Cambridge: Cambridge University Press, 2010. |
| [7] | H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, New York: Springer, 2 (2011). |
| [8] |
E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook Differ. Eq. Evol. Eq., 3 (2007), 527–621. https://doi.org/10.1016/S1874-5717(07)80010-7 doi: 10.1016/S1874-5717(07)80010-7
|
| [9] | A. Y. Khapalov, Controllability of partial differential equations governed by multiplicative controls, Heidelberg: Springer, 2010. |
| [10] |
P. Cannarsa, G. Floridia, A. Y. Khapalov, Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign, J. Math. Pure. Appl., 108 (2017), 425–458. https://doi.org/10.1016/j.matpur.2017.07.002 doi: 10.1016/j.matpur.2017.07.002
|
| [11] |
K. Ito, K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM J. Control Optim., 46 (2007), 274–287. https://doi.org/10.1137/05064254X doi: 10.1137/05064254X
|
| [12] |
E. Zerrik, M. O. Sidi, Regional controllability for infinite-dimensional bilinear systems: Approach and simulations, Int. J. Control, 84 (2011), 2108–2116. https://doi.org/10.1080/00207179.2011.632442 doi: 10.1080/00207179.2011.632442
|
| [13] | K. Ztot, E. Zerrik, H. Bourray, Regional control problem for distributed bilinear systems, Int. J. Ap. Mat. Com.-Pol., 21 (2011), 499–508. |
| [14] | S. Lenhart, J. T. Workman, Optimal control applied to biological models, Boca Raton: Chapman & Hall/CRC, 2007. |
| [15] |
R. Glowinski, Y. Song, X. Yuan, H. Yue, Bilinear optimal control of an advection-reaction-diffusion system, SIAM Rev., 64 (2022), 392–421. https://doi.org/10.1137/21M1389778 doi: 10.1137/21M1389778
|
| [16] | A. V. Fursikov, Optimal control of distributed systems. Theory and applications, Translations of Mathematical Monographs, American Mathematical Society, 1999. |
| [17] | R. Becker, B. Vexler, Optimal control of the convection–diffusion equation using stabilized finite element methods, Numer. Math., 106 (2007), 349–367. |
| [18] |
A. Kröner, B. Vexler, A priori error estimates for elliptic optimal control problems with a bilinear state equation, J. Comput. Appl. Math., 230 (2009), 781–802. https://doi.org/10.1016/j.cam.2009.01.023 doi: 10.1016/j.cam.2009.01.023
|
| [19] |
A. Borzì, E. J. Park, M. Vallejos Lass, Multigrid optimization methods for the optimal control of convection–diffusion problems with bilinear control, J. Optimiz. Theory App., 168 (2016), 510–533. https://doi.org/10.1007/s10957-015-0791-z doi: 10.1007/s10957-015-0791-z
|
| [20] |
A. Fleig, R. Guglielmi, Optimal control of the Fokker–Planck equation with space-dependent controls, J. Optimiz. Theory App., 174 (2017), 408–427. https://doi.org/10.1007/s10957-017-1120-5 doi: 10.1007/s10957-017-1120-5
|
| [21] | R. Glowinski, Variational methods for the numerical solution of nonlinear elliptic problems, Philadelphia: Society for Industrial and Applied Mathematics (SIAM), 2015. |
| [22] |
F. Bersetche, F. Fuica, E. Otárola, D. Quero, Bilinear optimal control for the fractional Laplacian: Analysis and discretization, SIAM J. Numer. Anal., 62 (2024), 1344–1371. https://doi.org/10.1137/23M154947X doi: 10.1137/23M154947X
|
| [23] |
C. Kenne, L. Djomegne, J. Larrouy, Bilinear optimal control of a one-dimensional degenerate parabolic equation with a nonlocal term, Math. Method. Appl. Sci., 47 (2024), 11670–11692. https://doi.org/10.1002/mma.10148 doi: 10.1002/mma.10148
|
| [24] |
F. Achab, A. Hafdallah, A. Ayadi, I. Rezzoug, Optimal control of a bilinear distributed system with incomplete data, Bound Value Probl., 2025 (2025), 159. https://doi.org/10.1186/s13661-025-02113-8 doi: 10.1186/s13661-025-02113-8
|
| [25] |
N. E. Boukhari, E. H. Zerrik, Optimal control of a class of bilinear systems with unbounded control operators: A semigroup approach, J. Control Decis., 2025 (2025), 1–13. https://doi.org/10.1080/23307706.2025.2566994 doi: 10.1080/23307706.2025.2566994
|
| [26] |
G. Mophou, Bilinear optimal control for a nonlinear parabolic equation with nonlocal time dependence, Axioms, 14 (2025), 38. https://doi.org/10.3390/axioms14010038 doi: 10.3390/axioms14010038
|
| [27] |
G. Mophou, C. Kenne, M. Warma, Bilinear optimal control for a fractional diffusive equation, Optimization, 75 (2026), 267–293. https://doi.org/10.1080/02331934.2024.2421384 doi: 10.1080/02331934.2024.2421384
|
| [28] |
E. Otárola, D. Quero, M. Sasso, A bilinear pointwise tracking optimal control problem for semilinear PDEs, J. Comput. Phys., 2026 (2026). https://doi.org/10.2139/ssrn.5499625 doi: 10.2139/ssrn.5499625
|
| [29] |
A. Albarrak, M. Benoudi, M. O. Sidi, I. O. Ahmed, H. O. Sidi, An inverse problem for recovering a time-dependent source in time fractional diffusion–wave models governed by the integral fractional Laplacian, Chaos Soliton. Fract., 205 (2026), 117887. https://doi.org/10.1016/j.chaos.2026.117887 doi: 10.1016/j.chaos.2026.117887
|
| [30] |
E. S. Baranovskii, R. V. Brizitskii, Z. Y. Saritskaia, Optimal control problems for the reaction–diffusion–convection equation with variable coefficients, Nonlinear Anal.-Real, 75 (2024), 103979. https://doi.org/10.1016/j.nonrwa.2023.103979 doi: 10.1016/j.nonrwa.2023.103979
|
| [31] |
M. Johnson, M. M. Raja, V. Vijayakumar, A. Shukla, Optimal control results for fractional differential hemivariational inequalities of order $r\in(1, 2)$, Optimization, 74 (2025), 1425–1449. https://doi.org/10.1080/02331934.2024.2306304 doi: 10.1080/02331934.2024.2306304
|
| [32] |
M. M. Raja, V. Vijayakumar, C.-W. Tsai, K. C. Veluvolu, Optimal control strategies and continuous dependence for stochastic Hilfer fractional systems with delay: A Volterra–Fredholm integro-differential approach, Optim. Contr. Appl. Met., 46 (2025), 2708–2726. https://doi.org/10.1002/oca.70024 doi: 10.1002/oca.70024
|
| [33] |
M. M. Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, N. Sakthivel, K. Kaliraj, Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order $r\in(1, 2)$, Optim. Contr. Appl. Met., 43 (2022), 996–1019. https://doi.org/10.1002/oca.2867 doi: 10.1002/oca.2867
|
| [34] |
N. I. Chaudhary, M. A. Z. Raja, Z. A. Khan, A. Mehmood, S. M. Shah, Design of fractional hierarchical gradient descent algorithm for parameter estimation of nonlinear control autoregressive systems, Chaos Soliton. Fract., 157 (2022), 111913. https://doi.org/10.1016/j.chaos.2022.111913 doi: 10.1016/j.chaos.2022.111913
|
| [35] |
N. I. Chaudhary, Z. A. Khan, A. K. Kiani, M. A. Z. Raja, I. I. Chaudhary, C. M. Pinto, Design of auxiliary model based normalized fractional gradient algorithm for nonlinear output-error systems, Chaos Soliton. Fract., 163 (2022), 112611. https://doi.org/10.1016/j.chaos.2022.112611 doi: 10.1016/j.chaos.2022.112611
|
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Mofareh Alhazmi, Maawiya Ould Sidi. Bilinear optimal control problem with minimum energy for a fourth-order reaction diffusion system[J]. AIMS Mathematics, 2026, 11(4): 10274-10310. doi: 10.3934/math.2026425
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